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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Bounded homotopy equivalences of Hilbert cube manifolds


Author: C. Bruce Hughes
Journal: Trans. Amer. Math. Soc. 287 (1985), 621-643
MSC: Primary 57N20; Secondary 18F25, 19D99, 19L99, 19M05, 55R65
DOI: https://doi.org/10.1090/S0002-9947-1985-0768729-X
MathSciNet review: 768729
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Abstract: Let $ M$ and $ F$ be Hilbert cube manifolds with $ F$ compact. The purpose of this paper is to study homotopy equivalences $ f:M \to {{\mathbf{R}}^m} \times F$ which have bounded control in the $ {{\mathbf{R}}^m}$-direction. Roughly, these homotopy equivalences form a semi-simplicial complex $ \mathcal{W}\mathcal{H}({{\mathbf{R}}^m} \times F)$, the controlled Whitehead space. Using results about approximate fibrations, $ \mathcal{W}\mathcal{H}({{\mathbf{R}}^m} \times F)$ is related to the semi-simplicial complex of bounded concordances on $ {{\mathbf{R}}^m} \times F$. Then the homotopy groups of $ \mathcal{W}\mathcal{H}({{\mathbf{R}}^m} \times F)$ are computed in terms of the lower algebraic $ K$-theoretic functors $ {K_{ - i}}$.


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DOI: https://doi.org/10.1090/S0002-9947-1985-0768729-X
Keywords: Approximate fibration, bounded homotopy equivalence, concordance, Hilbert cube manifold, lower algebraic $ K$-theory
Article copyright: © Copyright 1985 American Mathematical Society

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